We consider an atomic congestion game in which each player $i$ either participates in the game with an exogenous and known probability $p_{i}\in(0,1]$, independently of everybody else, or stays out and incurs no cost. We compute the parameterized price of anarchy to characterize the impact of demand uncertainty on the efficiency of selfish behavior, considering two different notions of a social planner. A prophet planner knows the realization of the random participation in the game; the ordinary planner does not. As a consequence, a prophet planner can compute an adaptive social optimum that selects different solutions depending on the players that turn out to be active, whereas an ordinary planner faces the same uncertainty as the players and can only compute social optima with respect to the player participation distribution. For both planners, we derive the precise price of anarchy, which arises from an optimization problem parameterized by the maximum participation probability $q=\max_{i} p_{i}$. In the case of affine costs, we provide an analytic expression for the ordinary and prophet price of anarchy, parameterized as a function of $q$.

翻译：我们考虑一种原子拥塞博弈，其中每位玩家$i$以独立于他人的外生已知概率$p_{i}\in(0,1]$参与博弈，否则不参与且不产生任何成本。我们计算参数化的无政府价格（price of anarchy），以刻画需求不确定性对自私行为效率的影响，并考虑两种不同的社会规划者概念。先知规划者知晓博弈中随机参与的实现结果；而普通规划者则不知晓。因此，先知规划者能够计算一种适应性社会最优解，该解根据实际活跃的玩家选择不同方案；而普通规划者面临与玩家相同的不确定性，只能基于玩家参与分布计算社会最优解。针对两种规划者，我们推导出精确的无政府价格，该价格源于一个由最大参与概率$q=\max_{i} p_{i}$参数化的优化问题。在仿射成本情形下，我们给出了普通与先知无政府价格的解析表达式，该表达式作为$q$的函数进行参数化。